To a local system of complex vector spaces on a topological space Chern and Simons associated natural characteristic classes living in odd-degree cohomology of the space with coefficients in C/Z. These classes refine the Chern classes of the complex vector bundle corresponding to the local system; the latter are always torsion, but Chern-Simons classes are generally non-trivial even rationally. For example, when evaluated on the uniformizing local system of a compact hyperbolic 3-manifold they recover the volume of the manifold.
I will discuss an analog of this theory for local systems of p-adic vector spaces. The p-adic theory turns out to exist not only for topological spaces but also for arithmetic objects such as algebraic varieties over number fields. While for local systems on complex algebraic varieties the characteristic classes of p-adic local systems are essentially always zero, for varieties over non-closed fields they are often non-trivial and in some cases behave tantalizingly similarly to characteristic classes of complex and real local systems; for example, one finds a p-adic analog of the Milnor-Wood inequality. This is joint work with Lue Pan.
427 Thackeray Hall